As to my knowledge you can't change the colour working space in Lightroom. When editing in LR you will edit in ProPhoto RGB, 16 bit. When editing in Photoshop in sRGB, 8 bit, you will loose colour. The best way is working in Photoshop in ProPhoto RGB 16 bit too. Only at the end of your editing, export the file in the colour space that's most appropriate, eg sRGB for use on a website.

Photoshop doesn't save automatically your changes, you have to save the file yourself after editing ;)

sRGB has a much smaller gamut than that of the ProPhotoRGB. That extra colors that Lightroom can handle in ProPhotoRGB will have to be squeezed and clipped when mapping to sRGB, resulting the loss of data. You could export the photo in full fidelity in ProPhotoRGB from Lightroom and have PS handle the ProPhotoRGB to sRGB conversion. That is a better workflow resulting in less data loss.

I think that is the default Lightroom user prefs under Preferences>External Editing>Edit in Photoshop